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Guidance

If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional. There are many theorems about triangles that you can prove using similar triangles.

Triangle Proportionality Theorem:
A line parallel to one side of a triangle divides the other two sides of the triangle proportionally.
This theorem and its converse will be explored and proved in Example A, Example B, and the practice exercises.

Triangle Angle Bisector Theorem:
The angle bisector of one angle of a triangle divides the opposite side of the triangle into segments proportional to the lengths of the other two sides of the triangle.
This theorem will be explored and proved in Example C.

Pythagorean Theorem:
For a right triangle with legs
and
and hypotenuse
,
.
This theorem will be explored and proved in the Guided Practice problems.

Example A

Prove that
.

Solution:
The two triangles share
. Because
, corresponding angles are congruent. Therefore,
. The two triangles have two pairs of congruent angles. Therefore,
by
.

Example B

Use your result from Example A to prove that
. Then, use algebra to show that
.

Solution:
which means that corresponding sides are proportional. Therefore,
. Now, you can use algebra to show that the second proportion must be true. Remember that
and
.

You have now
proved the triangle proportionality theorem:
a line parallel to one side of a triangle divides the other two sides of the triangle proportionally.

Example C

Consider
with
the angle bisector of
and point
constructed so that
. Prove that
.

Now all you need to show is that
in order to prove the desired result.

Because
is the angle bisector of
,
.

Because
,
(corresponding angles).

Because
,
(alternate interior angles).

Thus,
by the transitive property.

Therefore,
is isosceles because its base angles are congruent and it must be true that
. This means that
. Therefore:

This proves the triangle angle bisector theorem:
the angle bisector of one angle of a triangle divides the opposite side of the triangle into segments proportional to the lengths of the other two sides of the triangle.

Concept Problem Revisited

There are three triangles in this picture:
,
,
. All three triangles are right triangles so they have one set of congruent angles (the right angle).
and
share
, so
by
. Similarly,
and
share
, so
by
. By the transitive property, all three triangles must be similar to one another.